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The Eigenvalue Problem for the complex Monge-Ampère operator

Papa Badiane, Ahmed Zeriahi

Abstract

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Ampère operator on a bounded strongly pseudoconvex domain in $\C^n$. We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in $Ω$ and boundary values $0$. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P.L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge-Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck \cite{CKNS85} and B. Guan \cite{GuanB98}. Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge-Ampère operator.

The Eigenvalue Problem for the complex Monge-Ampère operator

Abstract

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Ampère operator on a bounded strongly pseudoconvex domain in . We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in and boundary values . Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P.L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge-Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck \cite{CKNS85} and B. Guan \cite{GuanB98}. Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge-Ampère operator.
Paper Structure (22 sections, 22 theorems, 236 equations)

This paper contains 22 sections, 22 theorems, 236 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. The complex Monge-Ampère opertaor
  4. The CKNS theorem
  5. The subsolution theorem of B. Guan
  6. The eigenvalue problem for linear elliptic operators
  7. A new existence theorem
  8. The gradient a priori estimate
  9. Laplacian estimate
  10. Boundary second order a priori estimates
  11. Global a priori estimates
  12. Proof of Theorem \ref{['thm:new']}
  13. The existence of the first eigenvalue
  14. Proof of Theorem \ref{['thm1']}
  15. An application
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\Omega \Subset \mathbb{C}^n$ be a bounded strongly pseudoconvex domain with smooth boundary and $0 < f \in C^{\infty}(\bar{\Omega})$ be a smooth positive function in $\bar{\Omega}$. Then there exists a real number $\lambda_1 = \lambda_1(\Omega,f)>0$ and a function $u_1 \in PSH(\Omega)\cap C^{\i

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Theorem 3.1
  • Proposition 3.2
  • ...and 13 more